### Büchi's problem in any power for finite fields

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Let Sq denote the set of squares, and let $S{Q}_{n}$ be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let ${B}_{n}(x,y)=\left(\genfrac{}{}{0pt}{}{x+y}{x}\right)MODn$. For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.

We prove that the positive-existential theory of addition and divisibility in a ring of polynomials in two variables A[t₁,t₂] over an integral domain A is undecidable and that the universal-existential theory of A[t₁] is undecidable.

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: if $K$ is recursive, then Hilbert’s Tenth Problem is undecidable in $R$. In general, there exist ${x}_{1},...,{x}_{n}\in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\mathbb{Q}({x}_{1},...,{x}_{n})$ has solutions in $R$.

We prove that Hilbert’s Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over $K$). We relate division-ample sets to arithmetic of abelian varieties.

We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the kth powers of the unknowns, with coefficients in C? We state a number-theoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = ℤ. We reduce a negative answer for k = 2 and for...

We show that it is decidable whether or not a given Q-rational series in several noncommutative variables has a cyclic image. By definition, a series r has a cyclic image if there is a rational number q such that all nonzero coefficients of r are integer powers of q.

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $\gamma \mapsto ord\left(Res\left({\phi}^{\gamma}\right)\right)$ factors through a function $ordRe{s}_{\phi}\left(\xb7\right)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P{\xb9}_{K}$, or on a segment, and the minimal resultant locus is contained in the tree in $P{\xb9}_{K}$ spanned by the fixed points and poles...

The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.